Equations of Motion of Systems with Internal Angular Momentum -II ManuelM. Dorado Dorado∗ CiRTA,ENITA, SL. S.A., Pol. Ind. Miguel Tres Yuste, Cantos12, Oeste, 28037 s/n Madrid, Spain Abstract Using the Euler’s equations and the Hamiltonian formulation, an attempt has been made to obtain the equations of motion of systems with internal angular momentum that are moving with respect to a reference frame when subjected to an interaction. This interaction involves the application of a torque that is permanently perpendicular to the internal angular momentum vector.
mass) L , whose centre of mass moves at a constant velocity ν with respect to a reference frame which INTRODUCTION canbedefinedassoonastheinteractioninthe cylinder starts. It is aimed to obtain the equations The angular momentum of a many-particle sys- of motion to describe the dynamical behaviour of tem with respect to its centre of mass is known as the system from the instant that it undergoes an M the ”internal angular momentum” and is a prop- interaction, by applying a torque that is perma- erty of the system that is independent of the ob- nently perpendicular to the internal angular mo- server. Internal angular momentum is therefore mentum (See Fig.1). and attribute that characterizes a system in the When solving the problem, the following points same way as its mass or charge. In the case of a are taken into account: rigid body and particularly in the case of an ele- (a) The cylinder will continue spinning about its mentary particle, the internal angular momentum longitudinal axis at a constant angular veloc- is also referred to as ”spin”. ity ω throughout all the movement. In other To the author’s knowledge, the dynamical be- words, its internal angular momentum mod- haviour of systems with internal angular momen- ule is constant. The energy that the cylinder tum has not been systematically studied within the possesses as a result of its rotation about its framework of classical mechanics (See References longitudinal axis is consequently considered to [1]). This paper approaches this type of problem be internal and does not interfere with its dy- through a model comprising a rotating cylinder, namical behaviour. with constant velocity of rotation, around its longi- tudinal axis. The equations of motion are obtained (b) The derivative of the internal angular momen- using the Euler’s equations and by the Hamiltonian tum, L , with respect to a frame of axes of iner- procedure. tial reference (X, Y , Z) satisfies the equation Results coincide when the problem is solved us- ing vectorial algebra or Lagrangian formalism (un- dL dL × L published). However, as a result of changes in ”per- dt = dt + Ω (1) spective”, each method uncovers new peculiarities XY Z X Y Z regarding the intimate nature of the system’s be- haviour. in which Ω is the rotation velocity of the frame linked to the solid (X , Y , Z )abouttheframe X Y Z FORMULATION OF THE PROBLEM of inertial reference axes ( , , ). (c) Only infinitesimal motions are considered that Let us consider a rigid cylindrical solid with internal are compatible with the system’s configura- angular momentum (with respect to its centre of tional variations brought about by the applied ∗ Email:E-mail: [email protected] [email protected] torque. (d) The total energy of the system is not explicitly (a) ω: the rotational velocity of the body around dependent on time. its longest axis (let us call it the intrinsic ro- tational velocity of the body). (e) No term is included that refers to the potential energy. According to hypothesis, the applied (b) Ω: the rotational velocity of the system of co- X Y torque is a null force acting on the system and ordinates associated with the body ( , , Z X Y the possibility of including a potential from ) viewed from the frame of inertia ( , , Z which the applied torque derives is unknown. ). We can write L and M , refered to (X , Y , Z ), as follows: